HOME

TheInfoList



OR:

The Stark effect is the shifting and splitting of
spectral line A spectral line is a dark or bright line in an otherwise uniform and continuous spectrum, resulting from emission or absorption of light in a narrow frequency range, compared with the nearby frequencies. Spectral lines are often used to iden ...
s of atoms and molecules due to the presence of an external
electric field An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field fo ...
. It is the electric-field analogue of the
Zeeman effect The Zeeman effect (; ) is the effect of splitting of a spectral line into several components in the presence of a static magnetic field. It is named after the Dutch physicist Pieter Zeeman, who discovered it in 1896 and received a Nobel prize ...
, where a spectral line is split into several components due to the presence of the
magnetic field A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to ...
. Although initially coined for the static case, it is also used in the wider context to describe the effect of time-dependent electric fields. In particular, the Stark effect is responsible for the
pressure broadening A spectral line is a dark or bright line in an otherwise uniform and continuous spectrum, resulting from emission (electromagnetic radiation), emission or absorption (electromagnetic radiation), absorption of light in a narrow frequency range, ...
(Stark broadening) of spectral lines by charged particles in
plasma Plasma or plasm may refer to: Science * Plasma (physics), one of the four fundamental states of matter * Plasma (mineral), a green translucent silica mineral * Quark–gluon plasma, a state of matter in quantum chromodynamics Biology * Blood pla ...
s. For most spectral lines, the Stark effect is either linear (proportional to the applied electric field) or quadratic with a high accuracy. The Stark effect can be observed both for emission and absorption lines. The latter is sometimes called the inverse Stark effect, but this term is no longer used in the modern literature.


History

The effect is named after the German physicist
Johannes Stark Johannes Stark (, 15 April 1874 – 21 June 1957) was a German physicist who was awarded the Nobel Prize in Physics in 1919 "for his discovery of the Doppler effect in canal rays and the splitting of spectral lines in electric fields". This phe ...
, who discovered it in 1913. It was independently discovered in the same year by the Italian physicist Antonino Lo Surdo, and in Italy it is thus sometimes called the Stark–Lo Surdo effect. The discovery of this effect contributed importantly to the development of quantum theory and Stark was awarded with the
Nobel Prize in Physics ) , image = Nobel Prize.png , alt = A golden medallion with an embossed image of a bearded man facing left in profile. To the left of the man is the text "ALFR•" then "NOBEL", and on the right, the text (smaller) "NAT•" then " ...
in the year 1919. Inspired by the magnetic
Zeeman effect The Zeeman effect (; ) is the effect of splitting of a spectral line into several components in the presence of a static magnetic field. It is named after the Dutch physicist Pieter Zeeman, who discovered it in 1896 and received a Nobel prize ...
, and especially by
Hendrik Lorentz Hendrik Antoon Lorentz (; 18 July 1853 – 4 February 1928) was a Dutch physicist who shared the 1902 Nobel Prize in Physics with Pieter Zeeman for the discovery and theoretical explanation of the Zeeman effect. He also derived the Lorentz t ...
's explanation of it,
Woldemar Voigt Woldemar Voigt (; 2 September 1850 – 13 December 1919) was a German physicist, who taught at the Georg August University of Göttingen. Voigt eventually went on to head the Mathematical Physics Department at Göttingen and was succeeded in 1 ...
performed classical mechanical calculations of quasi-elastically bound electrons in an electric field. By using experimental indices of refraction he gave an estimate of the Stark splittings. This estimate was a few orders of magnitude too low. Not deterred by this prediction, Stark undertook measurements on excited states of the hydrogen atom and succeeded in observing splittings. By the use of the Bohr–Sommerfeld ("old") quantum theory, Paul Epstein and
Karl Schwarzschild Karl Schwarzschild (; 9 October 1873 – 11 May 1916) was a German physicist and astronomer. Schwarzschild provided the first exact solution to the Einstein field equations of general relativity, for the limited case of a single spherical non-r ...
were independently able to derive equations for the linear and quadratic Stark effect in
hydrogen Hydrogen is the chemical element with the symbol H and atomic number 1. Hydrogen is the lightest element. At standard conditions hydrogen is a gas of diatomic molecules having the formula . It is colorless, odorless, tasteless, non-toxic, an ...
. Four years later,
Hendrik Kramers Hendrik Anthony "Hans" Kramers (17 December 1894 – 24 April 1952) was a Dutch physicist who worked with Niels Bohr to understand how electromagnetic waves interact with matter and made important contributions to quantum mechanics and statistical ...
derived formulas for intensities of spectral transitions. Kramers also included the effect of
fine structure In atomic physics, the fine structure describes the splitting of the spectral lines of atoms due to electron spin and relativistic corrections to the non-relativistic Schrödinger equation. It was first measured precisely for the hydrogen atom ...
, with corrections for relativistic kinetic energy and coupling between electron spin and orbital motion. The first quantum mechanical treatment (in the framework of
Werner Heisenberg Werner Karl Heisenberg () (5 December 1901 – 1 February 1976) was a German theoretical physicist and one of the main pioneers of the theory of quantum mechanics. He published his work in 1925 in a breakthrough paper. In the subsequent series ...
's
matrix mechanics Matrix mechanics is a formulation of quantum mechanics created by Werner Heisenberg, Max Born, and Pascual Jordan in 1925. It was the first conceptually autonomous and logically consistent formulation of quantum mechanics. Its account of quantum j ...
) was by
Wolfgang Pauli Wolfgang Ernst Pauli (; ; 25 April 1900 – 15 December 1958) was an Austrian theoretical physicist and one of the pioneers of quantum physics. In 1945, after having been nominated by Albert Einstein, Pauli received the Nobel Prize in Physics fo ...
.
Erwin Schrödinger Erwin Rudolf Josef Alexander Schrödinger (, ; ; 12 August 1887 – 4 January 1961), sometimes written as or , was a Nobel Prize-winning Austrian physicist with Irish citizenship who developed a number of fundamental results in quantum theory ...
discussed at length the Stark effect in his third paper on quantum theory (in which he introduced his perturbation theory), once in the manner of the 1916 work of Epstein (but generalized from the old to the new quantum theory) and once by his (first-order) perturbation approach. Finally, Epstein reconsideredP. S. Epstein, ''The Stark Effect from the Point of View of Schroedinger's Quantum Theory'', Physical Review, vol 28, pp. 695–710 (1926) the linear and quadratic Stark effect from the point of view of the new quantum theory. He derived equations for the line intensities which were a decided improvement over Kramers's results obtained by the old quantum theory. While the first-order-perturbation (linear) Stark effect in hydrogen is in agreement with both the old Bohr–Sommerfeld model and the quantum-mechanical theory of the atom, higher-order corrections are not. Measurements of the Stark effect under high field strengths confirmed the correctness of the new quantum theory.


Mechanism


Overview

An electric field pointing from left to right, for example, tends to pull nuclei to the right and electrons to the left. In another way of viewing it, if an electronic state has its electron disproportionately to the left, its energy is lowered, while if it has the electron disproportionately to the right, its energy is raised. Other things being equal, the effect of the electric field is greater for outer
electron shell In chemistry and atomic physics, an electron shell may be thought of as an orbit followed by electrons around an atom's nucleus. The closest shell to the nucleus is called the "1 shell" (also called the "K shell"), followed by the "2 shell" (o ...
s, because the electron is more distant from the nucleus, so it travels farther left and farther right. The Stark effect can lead to splitting of
degenerate energy level Degeneracy, degenerate, or degeneration may refer to: Arts and entertainment * ''Degenerate'' (album), a 2010 album by the British band Trigger the Bloodshed * Degenerate art, a term adopted in the 1920s by the Nazi Party in Germany to descri ...
s. For example, in the
Bohr model In atomic physics, the Bohr model or Rutherford–Bohr model, presented by Niels Bohr and Ernest Rutherford in 1913, is a system consisting of a small, dense nucleus surrounded by orbiting electrons—similar to the structure of the Solar Syste ...
, an electron has the same energy whether it is in the 2s state or any of the 2p states. However, in an electric field, there will be
hybrid orbitals In chemistry, orbital hybridisation (or hybridization) is the concept of mixing atomic orbitals to form new ''hybrid orbitals'' (with different energies, shapes, etc., than the component atomic orbitals) suitable for the pairing of electrons to f ...
(also called
quantum superposition Quantum superposition is a fundamental principle of quantum mechanics. It states that, much like waves in classical physics, any two (or more) quantum states can be added together ("superposed") and the result will be another valid quantum ...
s) of the 2s and 2p states where the electron tends to be to the left, which will acquire a lower energy, and other hybrid orbitals where the electron tends to be to the right, which will acquire a higher energy. Therefore, the formerly degenerate energy levels will split into slightly lower and slightly higher energy levels.


Multipole expansion

The Stark effect originates from the interaction between a
charge Charge or charged may refer to: Arts, entertainment, and media Films * '' Charge, Zero Emissions/Maximum Speed'', a 2011 documentary Music * ''Charge'' (David Ford album) * ''Charge'' (Machel Montano album) * ''Charge!!'', an album by The Aqu ...
distribution (atom or molecule) and an external
electric field An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field fo ...
. The interaction energy of a continuous charge distribution \rho(\mathbf), confined within a finite volume \mathcal, with an external
electrostatic potential Electrostatics is a branch of physics that studies electric charges at rest ( static electricity). Since classical times, it has been known that some materials, such as amber, attract lightweight particles after rubbing. The Greek word for ambe ...
\phi(\mathbf) is V_ = \int_\mathcal \rho(\mathbf) \phi(\mathbf) \, d^3 \mathbf r. This expression is valid classically and quantum-mechanically alike. If the potential varies weakly over the charge distribution, the
multipole expansion A multipole expansion is a mathematical series representing a function that depends on angles—usually the two angles used in the spherical coordinate system (the polar and azimuthal angles) for three-dimensional Euclidean space, \R^3. Similarly ...
converges fast, so only a few first terms give an accurate approximation. Namely, keeping only the zero- and first-order terms, \phi(\mathbf) \approx \phi(\mathbf) - \sum_^3 r_i F_i, where we introduced the electric field F_i \equiv - \left. \left(\frac \right)\_ and assumed the origin 0 to be somewhere within \mathcal. Therefore, the interaction becomes V_ \approx \phi(\mathbf) \int_\mathcal \rho(\mathbf) d^3r - \sum_^3 F_i \int_\mathcal \rho(\mathbf) r_i d^3r \equiv q \phi(\mathbf) - \sum_^3 \mu_i F_i = q \phi(\mathbf) - \boldsymbol \cdot \mathbf , where q and \mathbf are, respectively, the total charge (zero moment) and the dipole moment of the charge distribution. Classical macroscopic objects are usually neutral or quasi-neutral (q = 0), so the first, monopole, term in the expression above is identically zero. This is also the case for a neutral atom or molecule. However, for an
ion An ion () is an atom or molecule with a net electrical charge. The charge of an electron is considered to be negative by convention and this charge is equal and opposite to the charge of a proton, which is considered to be positive by conve ...
this is no longer true. Nevertheless, it is often justified to omit it in this case, too. Indeed, the Stark effect is observed in spectral lines, which are emitted when an electron "jumps" between two
bound state Bound or bounds may refer to: Mathematics * Bound variable * Upper and lower bounds, observed limits of mathematical functions Physics * Bound state, a particle that has a tendency to remain localized in one or more regions of space Geography *B ...
s. Since such a transition only alters the internal degrees of freedom of the radiator but not its charge, the effects of the monopole interaction on the initial and final states exactly cancel each other.


Perturbation theory

Turning now to quantum mechanics an atom or a molecule can be thought of as a collection of point charges (electrons and nuclei), so that the second definition of the dipole applies. The interaction of atom or molecule with a uniform external field is described by the operator V_ = - \mathbf\cdot \boldsymbol. This operator is used as a perturbation in first- and second-order
perturbation theory In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middle ...
to account for the first- and second-order Stark effect.


First order

Let the unperturbed atom or molecule be in a ''g''-fold degenerate state with orthonormal zeroth-order state functions \psi^0_1, \ldots, \psi^0_g . (Non-degeneracy is the special case ''g'' = 1). According to perturbation theory the first-order energies are the eigenvalues of the ''g'' × ''g'' matrix with general element (\mathbf_)_ = \langle \psi^0_k , V_ , \psi^0_l \rangle = -\mathbf\cdot \langle \psi^0_k , \boldsymbol , \psi^0_l \rangle, \qquad k,l=1,\ldots, g. If ''g'' = 1 (as is often the case for electronic states of molecules) the first-order energy becomes proportional to the expectation (average) value of the dipole operator \boldsymbol, E^ = -\mathbf\cdot \langle \psi^0_1 , \boldsymbol , \psi^0_1 \rangle = -\mathbf\cdot \langle \boldsymbol \rangle. Because the electric dipole moment is a vector (
tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tenso ...
of the first rank), the diagonal elements of the perturbation matrix Vint vanish between states with a certain parity. Atoms and molecules possessing inversion symmetry do not have a (permanent) dipole moment and hence do not show a linear Stark effect. In order to obtain a non-zero matrix Vint for systems with an inversion center it is necessary that some of the unperturbed functions \psi^0_i have opposite parity (obtain plus and minus under inversion), because only functions of opposite parity give non-vanishing matrix elements. Degenerate zeroth-order states of opposite parity occur for excited hydrogen-like (one-electron) atoms or Rydberg states. Neglecting fine-structure effects, such a state with the principal quantum number ''n'' is ''n''2-fold degenerate and n^2 = \sum_^ (2 \ell + 1), where \ell is the azimuthal (angular momentum) quantum number. For instance, the excited ''n'' = 4 state contains the following \ell states, 16 = 1 + 3 + 5 +7 \;\; \Longrightarrow\;\; n=4\;\text\; s\oplus p\oplus d\oplus f. The one-electron states with even \ell are even under parity, while those with odd \ell are odd under parity. Hence hydrogen-like atoms with ''n''>1 show first-order Stark effect. The first-order Stark effect occurs in rotational transitions of symmetric top molecules (but not for linear and asymmetric molecules). In first approximation a molecule may be seen as a rigid rotor. A symmetric top
rigid rotor In rotordynamics, the rigid rotor is a mechanical model of rotating systems. An arbitrary rigid rotor is a 3-dimensional rigid object, such as a top. To orient such an object in space requires three angles, known as Euler angles. A special rigi ...
has the unperturbed eigenstates , JKM \rangle = (D^J_)^* \quad\text\quad M,K= -J,-J+1,\dots,J with 2(2''J''+1)-fold degenerate energy for , K, > 0 and (2''J''+1)-fold degenerate energy for K=0. Here ''D''''J''''MK'' is an element of the
Wigner D-matrix The Wigner D-matrix is a unitary matrix in an irreducible representation of the groups SU(2) and SO(3). It was introduced in 1927 by Eugene Wigner, and plays a fundamental role in the quantum mechanical theory of angular momentum. The complex con ...
. The first-order perturbation matrix on basis of the unperturbed rigid rotor function is non-zero and can be diagonalized. This gives shifts and splittings in the rotational spectrum. Quantitative analysis of these Stark shift yields the permanent
electric dipole moment The electric dipole moment is a measure of the separation of positive and negative electrical charges within a system, that is, a measure of the system's overall polarity. The SI unit for electric dipole moment is the coulomb-meter (C⋅m). The ...
of the symmetric top molecule.


Second order

As stated, the quadratic Stark effect is described by second-order perturbation theory. The zeroth-order
eigenproblem In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted ...
H^ \psi^0_k = E^_k \psi^0_k, \quad k=0,1, \ldots, \quad E^_0 < E^_1 \le E^_2, \dots is assumed to be solved. The perturbation theory gives E^_k = \sum_ \frac \equiv -\frac \sum_^3 \alpha_ F_i F_j with the components of the polarizability tensor α defined by \alpha_ = -2\sum_ \frac. The energy ''E''(2) gives the quadratic Stark effect. Neglecting the
hyperfine structure In atomic physics, hyperfine structure is defined by small shifts in otherwise degenerate energy levels and the resulting splittings in those energy levels of atoms, molecules, and ions, due to electromagnetic multipole interaction between the nucl ...
(which is often justified — unless extremely weak electric fields are considered), the polarizability tensor of atoms is isotropic, \alpha_ \equiv \alpha_0 \delta_ \Longrightarrow E^ = -\frac \alpha_0 F^2. For some molecules this expression is a reasonable approximation, too. It is important to note that for the ground state \alpha_0 is ''always'' positive, i.e., the quadratic Stark shift is always negative.


Problems

The perturbative treatment of the Stark effect has some problems. In the presence of an electric field, states of atoms and molecules that were previously bound (
square-integrable In mathematics, a square-integrable function, also called a quadratically integrable function or L^2 function or square-summable function, is a real number, real- or complex number, complex-valued measurable function for which the integral of the s ...
), become formally (non-square-integrable)
resonance Resonance describes the phenomenon of increased amplitude that occurs when the frequency of an applied periodic force (or a Fourier component of it) is equal or close to a natural frequency of the system on which it acts. When an oscillatin ...
s of finite width. These resonances may decay in finite time via field ionization. For low lying states and not too strong fields the decay times are so long, however, that for all practical purposes the system can be regarded as bound. For highly excited states and/or very strong fields ionization may have to be accounted for. (See also the article on the
Rydberg atom A Rydberg atom is an excited atom with one or more electrons that have a very high principal quantum number, ''n''. The higher the value of ''n'', the farther the electron is from the nucleus, on average. Rydberg atoms have a number of peculia ...
).


Applications

The Stark effect is at the basis of the spectral shift measured for voltage-sensitive dyes used for imaging of the firing activity of neurons.


See also

*
Zeeman effect The Zeeman effect (; ) is the effect of splitting of a spectral line into several components in the presence of a static magnetic field. It is named after the Dutch physicist Pieter Zeeman, who discovered it in 1896 and received a Nobel prize ...
*
Autler–Townes effect In spectroscopy, the Autler–Townes effect (also known as AC Stark effect), is a dynamical Stark effect corresponding to the case when an oscillating electric field (e.g., that of a laser) is tuned in resonance (or close) to the transition frequen ...
*
Quantum-confined Stark effect The quantum-confined Stark effect (QCSE) describes the effect of an external electric field upon the light absorption spectrum or emission spectrum of a quantum well (QW). In the absence of an external electric field, electrons and holes within th ...
*
Stark spectroscopy Stark spectroscopy (sometimes known as electroabsorption/emission spectroscopy) is a form of spectroscopy based on the Stark effect. In brief, this technique makes use of the Stark effect (or electrochromism) either to reveal information about th ...
*
Inglis–Teller equation The Inglis–Teller equation represents an approximate relationship between the plasma density and the principal quantum number of the highest bound state of an atom. The equation was derived by David R. Inglis and Edward Teller in 1939. In a pl ...
* Electric field NMR * Stark effect in semiconductor optics


References


Further reading

* ''(Early history of the Stark effect)'' * ''(Chapter 17 provides a comprehensive treatment, as of 1935.)'' * ''(Stark effect for atoms)'' * ''(Stark effect for rotating molecules)'' {{DEFAULTSORT:Stark Effect Atomic physics Foundational quantum physics Physical phenomena Spectroscopy